Intermediate mathematics
See also: Category:Foundations A generalization (or generalisation) is the formulation of general concepts from specific instances by abstracting common properties. Generalization is the process of identifying the parts of a whole, as belonging to the whole. The parts, completely unrelated may be brought together as a group by establishing a common relation between them.Wikipedia:Generalization Foreword Mathematical notation can be extremely intimidating. Wikipedia is full of articles with page after page of indecipherable text. At first glance this article might appear to be the same. I want to assure the reader that every effort has been made to simplify everything as much as possible and to provide all relevant information or, at least, to make such information easy to find. Numbers :See: Peano axioms The basis of all of mathematics is the "Next" function (see Graph theory). Next(1)=2, Next(2)=3, Next(3)=4...This defines the Natural numbers (denoted \mathbb{N} ). These have the convenient property of being transitive. That means that if a1-3=x for which there is no answer. To provide an answer mathematicians generalize to the set of all integers (denoted \mathbb{Z} ) which includes negative integers and zero. Multiplication is defined as repeated addition, and its inverse is division. But this leads to equations like 3/2=x for which there is no answer. So mathematicians generalize to the set of rational numbers (denoted \mathbb{Q} ). Any number which isnt rational is irrational :(Addition and multiplication are fast but division is slow even for computers.) Exponentiation is defined as repeated multiplication, and its inverses are roots and logarithms. But this leads to multiple problems: :Equations like \sqrt{2}=x for which there is no answer so mathematicians generalize to the set of algebraic numbers (denoted \mathbb{A} ) ::Equations like 2^{\sqrt{2}}=x for which there is no answer (its transcendental) so mathematicians generalize to the set of Real numbers (denoted \mathbb{R} ). :Equations like \sqrt{-1}=x and e^x=-1 for which there is no answer so mathematicians generalize to the set of complex numbers (denoted \mathbb{C} ) by defining i = \sqrt{-1}. A single complex number z=a+bi consists of a real part a'' and an imaginary part ''bi. The Complex conjugate of a complex number z is \bar z=a-bi. (Not to be confused with the dual of a vector.) :It then follows that e^{i \pi}=-1 because e^{ix}=\cos x+i\sin x. See below. Tetration is defined as repeated exponentiation and its inverses are called super-root and super-logarithm. : Imaginary numbers (denoted \mathbb{I} ) often occur in equations involving change with respect to time. If friction is resistance to motion then imaginary friction would be resistance to change of motion wrt time. (In other words, imaginary friction would be mass.) In fact, in the equation for the Spacetime interval (given below), time itself is an imaginary quantity. Complex numbers can be used to represent and perform rotations but only in 2 dimensions. Hypercomplex numbers like quaternions (denoted \mathbb{H} ) are one way to generalize complex numbers to some (but not all) higher dimensions. Tensors, on the other hand, can be used in any number of dimensions to represent and perform rotations and other linear transformations. Rotations in n dimensions are called SO(n). See Graphical explanation of Tensor components. :Any affine transformation is equivalent to a linear transformation followed by a translation of the origin. (The origin is always a fixed point for any linear transformation.) A tensor is a multivector. Understanding how tensors work leads one to geometric algebra. An equation written in geometric algebra is much more intuitive than one written in matrix form. Clifford algebra generalizes geometric algebra to complex space. When a quantity, like the charge of a single electron, becomes so small that it is insignificant we, quite justifiably, treat it as though it were zero. A quantity that can be treated as though it were zero, even though it very definitely is not, is called infinitesimal. If q is a finite ( q \cdot 1 ) amount of charge then using Leibniz's notation dq would be an infinitesimal ( q \cdot 1/\infty ) amount of charge. See Differential Likewise when a quantity becomes so large that a regular finite quantity becomes insignificant then we call it infinite. We would say that the mass of the ocean is infinite ( M \cdot \infty ) . But compared to the mass of the Milky Way galaxy our ocean is insignificant. So we would say the mass of the Galaxy is doubly infinite ( M \cdot \infty^2 ) . Infinity and the infinitesimal are called Hyperreal numbers (denoted {}^*\mathbb{R} ). In every case hyperreals behave exactly like real numbers. For example, 2 \cdot \infty is exactly twice as big as \infty. Set theory \{ x \in \mathbb{R} : x>1 \} means the set of all x's that are members of the reals such that x is greater than one. See Set. \{1,2\}\cup\{2,3\}=\{1,2,3\} Union \{1,2\}\cap\{2,3\}=\{2\} Intersection \exists There exists at least one \forall For all Geometry The one dimensional number line can be generalized to a multidimensional Cartesian coordinate system thereby creating multidimensional math (i.e. geometry). : \mathbb{R}^3 is the Cartesian product \mathbb{R} \times \mathbb{R} \times \mathbb{R}. : \mathbb{C}^3 is the Cartesian product \mathbb{C} \times \mathbb{C} \times \mathbb{C} A vector space is a coordinate space with vector addition and scalar multiplication (multiplication of a vector and a scalar belonging to a field. Field is defined below). :If {\mathbf e_1} , {\mathbf e_2} , {\mathbf e_3} are orthogonal unit basis vectors). :and {\mathbf u} , {\mathbf v} , {\mathbf x} are arbitrary vectors then we can (and usually do) write: ::'' {\mathbf u} = {\mathbf u_1}{\mathbf e_1} + {\mathbf u_2}{\mathbf e_2} + {\mathbf u_3}{\mathbf e_3} '' ::'' {\mathbf v} = {\mathbf v_1}{\mathbf e_1} + {\mathbf v_2}{\mathbf e_2} + {\mathbf v_3}{\mathbf e_3} '' ::'' {\mathbf x} = {\mathbf x_1}{\mathbf e_1} + {\mathbf x_2}{\mathbf e_2} + {\mathbf x_3}{\mathbf e_3} '' :A module generalizes a vector space by allowing multiplication of a vector and a scalar belonging to a ring. Ring is defined below. Coordinate systems define the length of vectors parallel to one of the axes but leave all other lengths undefined. This concept of "length" which only works for certain vectors is generalized as the "norm" which works for all vectors. The norm of vector v is \|\mathbf{v}\|. :Taxicab metric (see [[Lp space|''L'p'' space]]) :: \|\mathbf{v}\| = v_1 + v_2 + v_3 :In Euclidean space rigid objects can rotate because the Euclidean norm doesnt depend on the choice of coordinate system. See proof of the Pythagorean theorem to the right. :: \|\mathbf{v}\| = \sqrt{v_1^2 + v_2^2 + v_3^2} :In Minkowski space the Spacetime interval is :: \|s\| = \sqrt{x^2 + y^2 + z^2 + (cti)^2} :In complex space the most common norm of an n dimensional vector is obtained by treating it as though it were a regular real valued 2n dimensional vector in Euclidean space :: \left\| \boldsymbol{z} \right\| = \sqrt{z_1 \bar z_1 + \cdots + z_n \bar z_n} A manifold \mathbf{M} is a type of topological space in which each point has an infinitely small neighbourhood that is homeomorphic to Euclidean space. A manifold is locally, but not globally, Euclidean. :A Tangent space \mathbf{T}_p \mathbf{M} is the set of all vectors tangent to \mathbf{M} at point p. :Informally, a tangent bundle \mathbf{TM} on a differentiable manifold \mathbf{M} (blue circle in image to the right) is obtained by joining all the tangent spaces (red lines) together in a smooth and non-overlapping manner (red cylinder).Wikipedia:Tangent bundle The tangent bundle always has twice as many dimensions as the original manifold. ::A fiber bundle is a generalization of a vector bundle which is a generalization of a tangent bundle. :The cotangent bundle (Dual bundle) of a differentiable manifold is obtained by joining all the cotangent spaces. Sections of that bundle are known as differential one-forms. Multiplication of vectors Multiplication can be generalized to allow for multiplication of vectors in 5 different ways: Dot product (a Scalar): \mathbf{u}\bullet\mathbf{v} = \|\mathbf{u}\|\ \|\mathbf{v}\|\cos(\theta) = u_1 v_1 + u_2 v_2 + u_3 v_3 : \mathbf{u}\bullet\mathbf{v} = \begin{bmatrix}u_1 \mathbf{e_1} \\ u_2 \mathbf{e_2} \\ u_3 \mathbf{e_3} \end{bmatrix} \begin{bmatrix}v_1 \mathbf{e_1} & v_2 \mathbf{e_2} & v_3 \mathbf{e_3} \end{bmatrix} = \begin{bmatrix}u_1 v_1 \mathbf{e_1}\mathbf{e_1} + u_2 v_2 \mathbf{e_2}\mathbf{e_2} + u_3 v_3 \mathbf{e_3}\mathbf{e_3} \end{bmatrix} :Strangely, only parallel components multiply. In Euclidean space \|\mathbf{v}\|^2 = \mathbf{v}\bullet\mathbf{v}= Q(\mathbf{x}). The dot product of a rank n tensor and a rank m tensor results in a rank n-m tensor. ::The dot product can be generalized to the bilinear form \beta(\mathbf{u,v}) = scalar and its associated quadratic form Q(\mathbf{x}) = \beta(\mathbf{x,x}). Two vectors are orthogonal if \beta(\mathbf{u,v}) = 0. A bilinear form is symmetric if \beta(\mathbf{u,v}) = \beta(\mathbf{v,u}) :::The bilinear form can be further generalized to the inner product (a sesquilinear form) \langle u,v\rangle=\overline{\langle v,u\rangle} ::::A Hilbert space is an inner product space that is also a Complete metric space (there are no points missing from it). Outer product (a tensor): \mathbf{u} \otimes \mathbf{v}. :As one would expect, every component of one vector multipies with every component of the other vector. : \begin{align}\mathbf{u} \otimes \mathbf{v} = \begin{bmatrix}u_1 \mathbf{e_1} \\ u_2 \mathbf{e_2} \\ u_3 \mathbf{e_3} \end{bmatrix} \begin{bmatrix}v_1 \mathbf{e_1} & v_2 \mathbf{e_2} & v_3 \mathbf{e_3} \end{bmatrix} = \begin{bmatrix}u_1 v_1 \mathbf{e_1} \mathbf{e_1} & u_1 v_2 \mathbf{e_1} \mathbf{e_2} & u_1 v_3 \mathbf{e_1} \mathbf{e_3} \\ u_2 v_1 \mathbf{e_2} \mathbf{e_1} & u_2 v_2 \mathbf{e_2} \mathbf{e_2} & u_2 v_3 \mathbf{e_2} \mathbf{e_3} \\ u_3 v_1 \mathbf{e_3} \mathbf{e_1} & u_3 v_2 \mathbf{e_3} \mathbf{e_2} & u_3 v_3 \mathbf{e_3} \mathbf{e_3} \end{bmatrix} \end{align} ::The Tensor product generalizes the outer product. The tensor product of a rank n tensor and a rank m tensor results in a rank n+m tensor. Wedge product (a simple bivector): \mathbf{u} \wedge \mathbf{v} = \mathbf{u} \otimes \mathbf{v} - \mathbf{v} \otimes \mathbf{u} = \overline{\mathbf{v}} :The wedge product is also called the exterior product (sometimes mistakenly called the outer product). The term "exterior" comes from the exterior product of two vectors not being a vector. Just as a vector has length and direction so a bivector has an area and an orientation. In three dimensions \mathbf{u} \wedge \mathbf{v} is a pseudovector and its dual is the cross product. \overline{\mathbf{u} \wedge \mathbf{v}} = \mathbf{u} \times \mathbf{v} : \mathbf{u} \wedge \mathbf{u} = 0 : ::The Matrix commutator generalizes the wedge product. :: A_2 = A_1A_2 - A_2A_1 Regressive product: \mathbf{u} \vee \mathbf{v} = \underline{\overline{\mathbf{u}} \wedge \overline{\mathbf{v}}} = \overline{\underline{\mathbf{u}} \wedge \underline{\mathbf{v}}} :The regressive product of two vectors is the dual of the wedge product of the duals of the two vectors. Geometric product (a multivector): \mathbf{u} \mathbf{v} = \mathbf{u} \bullet \mathbf{v} + \mathbf{u} \wedge \mathbf{v} : \mathbf{e_1} \mathbf{e_1} = \mathbf{e_{11}} = \mathbf{e_1} \bullet \mathbf{e_1} + \mathbf{e_1} \wedge \mathbf{e_1} = 1 + 0 = 1 : \mathbf{e_1} \mathbf{e_2} = \mathbf{e_{12}} = \mathbf{e_1} \bullet \mathbf{e_2} + \mathbf{e_1} \wedge \mathbf{e_2} = 0 + \mathbf{e_1} \wedge \mathbf{e_2} = \overline{\mathbf{e_2}} : ^2 Where \epsilon_{\mathbf{a}} is the signature of the vector. |} Integration :See also: Hyperreal number The '''integral (antiderivative) is a generalization of multiplication. :For example: an object dropped from point r1 to point r2 will release energy but the usual equation mass \cdot gravity \bullet (r_1 - r_2) = energy cant be used if the strength of gravity is itself a function of radius. The strength of gravity at r1 would be different than it is at r2. And in fact g® = 1/r^2 (See inverse-square law.) :However, the corresponding Definite integral is easily solved: mass \cdot \int_{r_1}^{r_2} g® \cdot dr : The derivative is a generalization of division. The derivative of the integral of f(x) is just f(x). If we know the value of a continuous function and all its derivatives at point a'' then we can determine the value at any other point ''x by using the Taylor series. ("!" Is factorial) : f(a) + {(x-a)^1 \over 1!} f'(a) + {(x-a)^2 \over 2!}f''(a) + {(x-a)^3 \over 3!}f^{(3)}(a) + \cdots ::The Laurent series generalizes the Taylor series. See Cauchy's integral formula. Using a=0 we can easily determine the Taylor series expansion of the exponential function e^x . : e^x = \sum_{n = 0}^{\infty} {x^n \over n!} = {x^0 \over 0!} + {x^1 \over 1!} + {x^2 \over 2!} + {x^3 \over 3!} + {x^4 \over 4!} + \cdots And cos(x) and sin(x) : \cos x = \frac{x^0}{0!} - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots : \sin x = \frac{x^1}{1!} - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots The Taylor series cant be used for a discontinuous function like a square wave because it is not differentiable but remarkably we can use the Fourier series to expand it or any other periodic function of period p'' (and frequency f) into an infinite sum of sine waves (of frequency nf) each of which is an infinite sum of terms each one of which is differentiable! : s(t) = \frac{a_0}{2} + \sum_{n=1}^\infty \lefta_n\cos\left(nt\right)+b_n\sin\left(nt\right)\right : a_n = \frac{2}{P}\int_{t_0}^{t_0+P} s(t)\cdot \cos\left(\tfrac{2\pi nt}{P}\right)\ dt : b_n = \frac{2}{P}\int_{t_0}^{t_0+P} s(t)\cdot \sin\left(\tfrac{2\pi nt}{P}\right)\ dt Fourier transforms generalize Fourier series to nonperiodic functions. Functions that are localized in the time domain have Fourier transforms that are spread out across the frequency domain and vice versa, a phenomenon known as the uncertainty principle. is a typical transient response]] :Laplace transforms generalize Fourier transforms to complex frequency. Complex frequency includes a term corresponding to the amount of damping. ::Integral transforms generalize Laplace transforms to other kernals (besides sine and cosine) 'Partial derivatives''' and multiple integrals generalize derivatives and integrals to multiple dimensions. The partial derivative with respect to one variable \frac{\part f(x,y)}{\part x} is found by simply treating all other variables as though they were constants. Multiple integrals are found the same way. : The Lie derivative generalizes the Lie bracket which generalizes the wedge product which is a generalization of the cross product which only works in 3 dimensions. The cross product is neither commutative nor associative and therefore doesnt form a field or even a ring (see below). Instead it forms a Lie algebra (See Infinitesimal transformation) which is a local or linearized version of a Lie group. A Lie group is a group that is also a differentiable manifold. Generalization of addition and multiplication :Main article: Algebraic structure Addition and multiplication can be generalized in so many ways that mathematicians have created a whole system of categories just to organize them. Category theory :See also: Higher category theory A category is an algebraic structure similar to a group but without requiring inverse or closure properties Categories consist of: :Objects (usually Sets) :Morphisms (usually maps) possessing: ::one source object (domain) ::one target object (codomain) a morphism is represented by an arrow: : f(x)=y is written f : x \to y : g(y)=z is written g : y \to z The image of x is y. The preimage (or fiber) of y is the set of all x whose image is y and is denoted f^{-1}y A space Y is a covering space of space Z if the map g of Y to Z is locally homeomorphic. :A covering space is a universal covering space if it is simply connected. Composition of morphisms: : g(f(x)) is written g \circ f = h : x \to z Given a covering space g : y \to z, a Lift of map h : x \to z is any map f : x \to y for which g \circ f = h A homomorphism is a map from one set to another of the same type which preserves the operations of the algebraic structure: : f(x \bullet y) = f(x) \bullet f(y) ::A Functor is a homomorphism with a domain in one category and a codomain in another. A Multicategory has morphisms with more than one source object. A Multilinear map f(v_1,\ldots,v_n) = W : : f\colon V_1 \times \cdots \times V_n \to W\text{,} has a corresponding Linear map: F(v_1\otimes \cdots \otimes v_n) = W : : F\colon V_1 \otimes \cdots \otimes V_n \to W\text{,} A topological space is (path-)connected if no part of it is disconnected. A space is simply connected if there are no holes passing all the way through it (therefore any loop can be shrunk to a point) Dot product might not be as strange as it first appears :See Hodge dual Let \mathbf{v} be a covector (a pseudovector) that is the orthogonal complement of vector \mathbf{v}. (A covector would be a tensor of rank -1.) \mathbf{u} \bullet \mathbf{v} = \mathbf{u} \wedge \overline{\mathbf{v}}I = \begin{bmatrix} u_1 \mathbf{e_1} \\ u_2 \mathbf{e_2} \\ u_3 \mathbf{e_3} \end{bmatrix} \wedge \begin{bmatrix} v_1 \mathbf{e_2 e_3} & v_2 \mathbf{e_3 e_1} & v_3 \mathbf{e_1 e_2} \end{bmatrix} I Therefore: \mathbf{u} \bullet \mathbf{v} = \begin{bmatrix} u_1 v_1 \mathbf{e_1 \wedge e_2 e_3} & u_1 v_2 \mathbf{e_1 \wedge e_3 e_1} & u_1 v_3 \mathbf{e_1 \wedge e_1 e_2} \\ u_2 v_1 \mathbf{e_2 \wedge e_2 e_3} & u_2 v_2 \mathbf{e_2 \wedge e_3 e_1} & u_2 v_3 \mathbf{e_2 \wedge e_1 e_2} \\ u_3 v_1 \mathbf{e_3 \wedge e_2 e_3} & u_3 v_2 \mathbf{e_3 \wedge e_3 e_1} & u_3 v_3 \mathbf{e_3 \wedge e_1 e_2} \end{bmatrix} I Which reduces to: \mathbf{u} \bullet \mathbf{v} = \begin{bmatrix} u_1 v_1 \mathbf{e_{123}} & 0 & 0 \\ 0 & u_2 v_2 \mathbf{e_{123}} & 0 \\ 0 & 0 & u_3 v_3 \mathbf{e_{123}} \end{bmatrix} I = (u_1 v_1 + u_2 v_2 + u_3 v_3) Because a trivector in 3 dimensions is a pseudoscalar: : (\mathbf{e_{123}})I = 1 ::Trivector with unit volume. : (\mathbf{e_{223}})I = 0 ::Trivector with zero volume (since its 2 dimensional). So, just like the cross product, the dot product is the dual of the wedge product. But whereas the cross product is between two vectors the dot product is between a vector and a covector. The Regressive product is between 2 covectors. External links *http://mathinsight.org *https://math.stackexchange.com References This article incorporates text from Wikipedia:Category (mathematics) Category:Foundations